Evaluating limits.

Evaluate the limit, if it exists. (If it does not exist, enter NONE).

I know according to rule 5 of the law of limits, this cant work by substition because the denominator would equal 0. I know i need to start this off by factoring but im not sure how to start this process.

Remember that when x= a makes polynomial P(x) zero- that is if P(a)= 0, then x-a must be a factor of P(x). Here, you saw correctly that the denominator is 0 when x= 2. That should make you look immediately at the numerator!

If the numerator were NOT 0 at x= 2, that would tell you that the limit does not exist and you can stop.

If the numerator is 0 at x= 2, which is the case here, , you know that you can factor out x-2. Once you know that, it should be clear that .

Now you can think as long as .

Notice that "as long as ". Of course, is "undefined" at x= 2 while x+3 is equal to 5 at x= 2 so we cannot just say that . Fortunately there is an (often overlooked) theorem about limits that says "If f(x)= g(x) for all x except x=a, then ".

Knowing that for all x except x= 2 is enough to tell us they have the same limit at x= 2 and, of course, is easy.